Journal Article

(\\textsf {AD}^{+}\\) implies \\( \\omega _{1}\\) is a club \\( \\Theta \\)-Berkeley cardinal

Sargsyan, Grigor, Blue, Douglas

2025

Overview

Following [1], given cardinals \\(\\kappa <\\lambda \\), we say \\(\\kappa \\) is a club \\(\\lambda \\)-Berkeley cardinal if for every transitive set N of size \\(<\\lambda \\) such that \\(\\kappa \\subseteq N\\), there is a club \\(C\\subseteq \\kappa \\) with the property that for every \\(\\eta \\in C\\), there is an elementary embedding \\(j: N\\rightarrow N\\) with \\(\\mathrm {crit }(j)=\\eta \\). We say \\(\\kappa \\) is \\(\\nu \\)-club \\(\\lambda \\)-Berkeley if \\(C\\subseteq \\kappa \\) as above is a \\(\\nu \\)-club. We say \\(\\kappa \\) is \\(\\lambda \\)-Berkeley if C is unbounded in \\(\\kappa \\). We show that under \\(\\textsf {AD}^{+}\\), (1) every regular Suslin cardinal is \\(\\omega \\)-club \\(\\Theta \\)-Berkeley (see Theorem 7.1), (2) \\(\\omega _1\\) is club \\(\\Theta \\)-Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are \\(\\Theta \\)-Berkeley – in particular, \\(\\omega _2\\) is \\(\\Theta \\)-Berkeley (see Remark 7.5).Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming \\(V=L({\\mathbb {R}})+{\\textsf {AD}}\\), \\(\\omega _1\\) is not \\(\\Theta ^+\\)-Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).

• Publisher: Cambridge University Press
• Subject: Theorems